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import OAI.Probability.GaussianRegression.Main
import OAI.Probability.MemoryPrecision.MeasurableTests
namespace OAI
open MeasureTheory ProbabilityTheory Set Filter EuclideanGeometry
open scoped ENNReal NNReal RealInnerProductSpace Topology Pointwise
noncomputable section
universe u v
open MeasureTheory ProbabilityTheory Set Filter
open scoped ENNReal NNReal RealInnerProductSpace Topology Pointwise
namespace MemoryPrecision
open NoiselessRegression
def realNormalize {d : ℕ} (ν : Measure (Space d)) (β H : ℝ) (z : Space d) (r : ℝ) :
Measure (Space d) :=
(ENNReal.ofReal (H * r^β))⁻¹ •
(ν.restrict (Metric.closedBall z r)).map (fun x => r⁻¹ • (x-z))
lemma realNormalize_local {d : ℕ} {ν : Measure (Space d)} {β H : ℝ}
(hν : AmbientBallBound ν H β) (hH : 0 < H) (z : Space d) {r : ℝ} (hr : 0 < r) :
AmbientBallBound (realNormalize ν β H z r) 1 β := by
have hc0 : ENNReal.ofReal (H*r^β) ≠ 0 := by positivity
intro t ρ hρ
rw [realNormalize, Measure.smul_apply, smul_eq_mul,
Measure.map_apply (by fun_prop) Metric.isClosed_closedBall.measurableSet]
have hsub : (fun x : Space d => r⁻¹ • (x-z)) ⁻¹' Metric.closedBall t ρ ⊆
Metric.closedBall (z + r • t) (r*ρ) := by
intro x hx
have he : x-(z+r•t) = r • (r⁻¹ • (x-z)-t) := by
rw [smul_sub, smul_smul, mul_inv_cancel₀ hr.ne', one_smul]
abel
rw [Metric.mem_closedBall, dist_eq_norm, he, norm_smul, Real.norm_eq_abs, abs_of_pos hr]
exact mul_le_mul_of_nonneg_left
(by simpa only [Set.mem_preimage, Metric.mem_closedBall, dist_eq_norm] using hx) hr.le
calc
_ ≤ (ENNReal.ofReal (H*r^β))⁻¹ * ν (Metric.closedBall (z+r•t) (r*ρ)) :=
mul_le_mul' le_rfl ((Measure.restrict_le_self _).trans (measure_mono hsub))
_ ≤ (ENNReal.ofReal (H*r^β))⁻¹ * ENNReal.ofReal (H*(r*ρ)^β) :=
mul_le_mul' le_rfl (hν _ _ (mul_pos hr hρ))
_ = ENNReal.ofReal (1*ρ^β) := by
rw [Real.mul_rpow hr.le hρ.le, ← mul_assoc,
ENNReal.ofReal_mul (by positivity : 0 ≤ H*r^β), ← mul_assoc,
ENNReal.inv_mul_cancel hc0 ENNReal.ofReal_ne_top, one_mul, one_mul]
lemma realNormalize_support {d : ℕ} (ν : Measure (Space d)) (β H : ℝ)
(z : Space d) {r : ℝ} (hr : 0 < r) :
(realNormalize ν β H z r).support ⊆ Metric.closedBall 0 1 := by
apply Measure.support_subset_of_isClosed Metric.isClosed_closedBall
rw [mem_ae_iff, realNormalize, Measure.smul_apply, Measure.map_apply (by fun_prop)
Metric.isClosed_closedBall.measurableSet.compl,
Measure.restrict_apply ((by fun_prop : Measurable (fun x : Space d => r⁻¹ • (x-z)))
Metric.isClosed_closedBall.measurableSet.compl)]
have he : (fun x : Space d => r⁻¹ • (x-z)) ⁻¹' (Metric.closedBall 0 1)ᶜ ∩
Metric.closedBall z r = ∅ := by
apply Set.eq_empty_iff_forall_notMem.mpr
intro x ⟨hx,hb⟩
apply hx
rw [Metric.mem_closedBall, dist_zero_right, norm_smul, Real.norm_eq_abs,
abs_of_pos (inv_pos.mpr hr)]
exact (inv_mul_le_iff₀ hr).2 (by simpa [dist_eq_norm] using hb)
rw [he, measure_empty, smul_zero]
lemma realNormalize_finite {d : ℕ} (ν : Measure (Space d)) [IsFiniteMeasure ν]
{β H : ℝ} (hH : 0 < H) (z : Space d) {r : ℝ} (hr : 0 < r) :
IsFiniteMeasure (realNormalize ν β H z r) := by
constructor
rw [realNormalize, Measure.smul_apply, smul_eq_mul]
apply ENNReal.mul_lt_top
· exact ENNReal.inv_lt_top.mpr (by positivity)
· exact measure_lt_top _ _
lemma realNormalize_reconstruct {d : ℕ} (ν : Measure (Space d)) {β H : ℝ}
(hH : 0 < H) (z : Space d) {r : ℝ} (hr : 0 < r) :
ENNReal.ofReal (H*r^β) • (realNormalize ν β H z r).map (fun t => z+r•t) =
ν.restrict (Metric.closedBall z r) := by
have hc0 : ENNReal.ofReal (H*r^β) ≠ 0 := by positivity
rw [realNormalize, Measure.map_smul _ (by fun_prop), smul_smul,
ENNReal.mul_inv_cancel hc0 ENNReal.ofReal_ne_top, one_smul,
Measure.map_map (by fun_prop) (by fun_prop)]
have he : (fun t : Space d => z+r•t) ∘ (fun x => r⁻¹ • (x-z)) = id := by
ext x
simp [smul_smul, hr.ne']
rw [he, Measure.map_id]
lemma normalized_scale_le {d m p : ℕ} (hp : 2 ≤ p) {β K : ℝ} (hK : 1 ≤ K)
(hgap : (m:ℝ)+(p:ℝ)-2 < β)
(hJ : ∀ j : ℕ, j ≤ p-2 → J β m j ≤ K^d) :
ProjectionNormScale m p 1 β 1 ≤ K^d := by
have hp0 : (0:ℝ) < p := by exact_mod_cast (by omega : 0 < p)
have hprod0 : 0 ≤ ∏ j ∈ Finset.range (p-1), J β m j := by
apply Finset.prod_nonneg
intro j hj
apply (J_pos ?_).le
have hj' : (j:ℝ)+2 ≤ p := by exact_mod_cast (show j+2 ≤ p by have := Finset.mem_range.mp hj; omega)
linarith
have hprod : (∏ j ∈ Finset.range (p-1), J β m j) ≤ (K^d)^(p-1) := by
calc
_ ≤ ∏ _j ∈ Finset.range (p-1), K^d := by
apply Finset.prod_le_prod₀
· intro j hj
apply (J_pos ?_).le
have hj' : (j:ℝ)+2 ≤ p := by exact_mod_cast (show j+2 ≤ p by have := Finset.mem_range.mp hj; omega)
linarith
· intro j hj
exact hJ j (by have := Finset.mem_range.mp hj; omega)
_ = _ := by simp
have hpeak : (2*Real.pi) ^ (-(m:ℝ)*((p:ℝ)-1)/(2*(p:ℝ))) ≤ 1 := by
apply Real.rpow_le_one_of_one_le_of_nonpos
· linarith [Real.pi_gt_three]
· have : (1:ℝ) ≤ p := by exact_mod_cast (by omega : 1 ≤ p)
apply div_nonpos_of_nonpos_of_nonneg
· exact mul_nonpos_of_nonpos_of_nonneg (neg_nonpos.mpr (Nat.cast_nonneg m)) (by linarith)
· positivity
unfold ProjectionNormScale
simp only [mul_one, Real.one_rpow]
calc
_ ≤ 1 * (∏ j ∈ Finset.range (p-1), J β m j)^(1/(p:ℝ)) :=
mul_le_mul_of_nonneg_right hpeak (Real.rpow_nonneg hprod0 _)
_ ≤ ((K^d)^(p-1))^(1/(p:ℝ)) := by rw [one_mul]; exact Real.rpow_le_rpow hprod0 hprod (by positivity)
_ = (K^d)^(((p-1:ℕ):ℝ)/(p:ℝ)) := by
rw [← Real.rpow_natCast_mul (by positivity : 0 ≤ K^d)]
congr 1
ring
_ ≤ (K^d)^(1:ℝ) := by
apply Real.rpow_le_rpow_of_exponent_le (one_le_pow₀ hK)
rw [div_le_one hp0]
exact_mod_cast Nat.sub_le p 1
_ = _ := Real.rpow_one _
lemma normalized_weighted_selection {d m p N : ℕ} (ν : Fin N → Measure (Space d))
[∀ w, IsFiniteMeasure (ν w)] (hp : 2 ≤ p) (hm : 1 ≤ m) (hmd : m ≤ d)
{β K : ℝ} (hK : 1 ≤ K) (hgap : (m:ℝ)+(p:ℝ)-2 < β)
(hJ : ∀ j : ℕ, j ≤ p-2 → J β m j ≤ K^d)
(hball : ∀ w, AmbientBallBound (ν w) 1 β)
(hsupport : ∀ w, (ν w).support ⊆ Metric.closedBall 0 1)
(g : Fin N → ((Fin m → Space d) × (Fin m → ℝ)) → ℝ≥0∞)
(hg : ∀ w, Measurable (g w)) (hgsum : ∀ x, ∑ w, g w x ≤ 1) :
(∑ w, ∫⁻ x, g w x ∂projectionLaw (ν w) m) ≤
ENNReal.ofReal (K*192)^d * (N:ℝ≥0∞)^(1/(p:ℝ)) := by
classical
obtain ⟨D,hD,hDs,hDV⟩ := exists_projection_support (E := Space d) m (by simpa using hmd)
have hg1 (w) (x) : g w x ≤ 1 :=
(Finset.single_le_sum (fun _ _ => zero_le) (Finset.mem_univ w)).trans (hgsum x)
have hpoint (w) : (∫⁻ x, g w x ∂projectionLaw (ν w) m) ≤
ENNReal.ofReal (K^d) * (∫⁻ x in D, g w x ∂matrixLabelReference m)^(1-1/(p:ℝ)) := by
obtain ⟨F,hF,hF0,hden,_,hnorm⟩ := exact_finite_density d m p hp hm β 1 1 hgap
(by norm_num) (by norm_num) 0 (ν w) (hsupport w) (hball w)
have hFD : ∀ᵐ x ∂matrixLabelReference m, x ∉ D → F x = 0 := by
have hz : projectionLaw (ν w) m Dᶜ = 0 := by
rw [← compl_mem_ae_iff, compl_compl]
unfold projectionLaw
apply (ae_map_iff (by fun_prop) hD).2
filter_upwards [Measure.quasiMeasurePreserving_snd.ae (ν w).support_mem_ae] with x hx
exact hDs x.1 x.2 (by simpa using hsupport w hx)
change (matrixLabelReference m).withDensity _ = projectionLaw (ν w) m at hden
rw [← hden, withDensity_apply _ hD.compl] at hz
have ha := (lintegral_eq_zero_iff' hF.ennreal_ofReal.aemeasurable).mp hz
change ∀ᵐ x ∂(matrixLabelReference m).restrict Dᶜ, ENNReal.ofReal (F x) = 0 at ha
rw [ae_restrict_iff' hD.compl] at ha
filter_upwards [ha] with x hx hnot
exact le_antisymm (ENNReal.ofReal_eq_zero.mp (hx hnot)) (hF0 x)
have hn : eLpNorm F (p:ℝ≥0∞) (matrixLabelReference m) ≤ ENNReal.ofReal (K^d) :=
hnorm.trans (ENNReal.ofReal_le_ofReal (normalized_scale_le hp hK hgap hJ))
have ht := density_test_bound (matrixLabelReference m) F hF hF0 p hp
(ENNReal.ofReal (K^d)) hn D hD hFD (g w) (hg w).aemeasurable (hg1 w)
change (matrixLabelReference m).withDensity _ = projectionLaw (ν w) m at hden
rwa [hden] at ht
have hq0 : 0 ≤ 1-1/(p:ℝ) := by
have hp1 : (1:ℝ) ≤ p := by exact_mod_cast (by omega : 1 ≤ p)
linarith [one_div_le_one_div_of_le (by norm_num : (0:ℝ)<1) hp1]
have hsum : (∑ w, ∫⁻ x in D, g w x ∂matrixLabelReference m) ≤
matrixLabelReference (E := Space d) m D := by
rw [← lintegral_finsetSum _ (fun w _ => hg w)]
simpa using lintegral_mono (μ := (matrixLabelReference (E := Space d) m).restrict D) hgsum
have hv : (matrixLabelReference (E := Space d) m D)^(1-1/(p:ℝ)) ≤ (192:ℝ≥0∞)^d := by
calc
_ ≤ ((192:ℝ≥0∞)^d)^(1-1/(p:ℝ)) := ENNReal.rpow_le_rpow (by simpa using hDV) hq0
_ ≤ ((192:ℝ≥0∞)^d)^(1:ℝ) :=
ENNReal.rpow_le_rpow_of_exponent_le (one_le_pow₀ (by norm_num)) (sub_le_self _ (by positivity))
_ = _ := ENNReal.rpow_one _
calc
_ ≤ ∑ w, ENNReal.ofReal (K^d) * (∫⁻ x in D, g w x ∂matrixLabelReference m)^(1-1/(p:ℝ)) :=
Finset.sum_le_sum (fun w _ => hpoint w)
_ = ENNReal.ofReal (K^d) * ∑ w, (∫⁻ x in D, g w x ∂matrixLabelReference m)^(1-1/(p:ℝ)) := by rw [Finset.mul_sum]
_ ≤ ENNReal.ofReal (K^d) * ((N:ℝ≥0∞)^(1/(p:ℝ)) *
(∑ w, ∫⁻ x in D, g w x ∂matrixLabelReference m)^(1-1/(p:ℝ))) := by
apply mul_le_mul' le_rfl
simpa using finite_rpow_holder (fun w => ∫⁻ x in D, g w x ∂matrixLabelReference m)
(by exact_mod_cast (by omega : 1 < p))
_ ≤ ENNReal.ofReal (K^d) * ((N:ℝ≥0∞)^(1/(p:ℝ)) * (192:ℝ≥0∞)^d) :=
mul_le_mul' le_rfl (mul_le_mul' le_rfl ((ENNReal.rpow_le_rpow hsum hq0).trans hv))
_ = _ := by
rw [ENNReal.ofReal_pow (by linarith : 0 ≤ K), mul_left_comm, ← mul_pow,
ENNReal.ofReal_mul (by linarith : 0 ≤ K)]
norm_num
ring
lemma localized_weighted_selection {d m p N : ℕ} (ν : Fin N → Measure (Space d))
[∀ w, IsFiniteMeasure (ν w)] (hp : 2 ≤ p) (hm : 1 ≤ m) (hmd : m ≤ d)
{β K H : ℝ} (hK : 1 ≤ K) (hH : 0 < H) (hgap : (m:ℝ)+(p:ℝ)-2 < β)
(hJ : ∀ j : ℕ, j ≤ p-2 → J β m j ≤ K^d)
(hball : ∀ w, AmbientBallBound (ν w) H β)
(g : Fin N → ((Fin m → Space d) × (Fin m → ℝ)) → ℝ≥0∞)
(hg : ∀ w, Measurable (g w)) (hgsum : ∀ x, ∑ w, g w x ≤ 1)
(z : Space d) {r : ℝ} (hr : 0 < r) :
(∑ w, ∫⁻ x, g w x ∂projectionLaw ((ν w).restrict (Metric.closedBall z r)) m) ≤
ENNReal.ofReal ((K*192)^d * (N:ℝ)^(1/(p:ℝ)) * H * r^β) := by
let τ w := realNormalize (ν w) β H z r
let (w : Fin N) : IsFiniteMeasure (τ w) := realNormalize_finite (ν w) hH z hr
let F : ((Fin m → Space d) × (Fin m → ℝ)) → ((Fin m → Space d) × (Fin m → ℝ)) :=
fun x => (x.1, fun i => ⟪z,x.1 i⟫ + r*x.2 i)
have hF : Measurable F := by fun_prop
have hb := normalized_weighted_selection τ hp hm hmd hK hgap hJ
(fun w => realNormalize_local (hball w) hH z hr)
(fun w => realNormalize_support (ν w) β H z hr)
(fun w => g w ∘ F) (fun w => (hg w).comp hF) (fun x => hgsum (F x))
have he (w) : (∫⁻ x, g w x ∂projectionLaw ((ν w).restrict (Metric.closedBall z r)) m) =
ENNReal.ofReal (H*r^β) * ∫⁻ x, g w (F x) ∂projectionLaw (τ w) m := by
rw [← realNormalize_reconstruct (ν w) (β := β) hH z hr, projectionLaw_smul,
projectionLaw_affine, lintegral_smul_measure, smul_eq_mul, lintegral_map (hg w) hF]
simp_rw [he]
rw [← Finset.mul_sum]
refine (mul_le_mul' le_rfl hb).trans_eq ?_
rw [← ENNReal.ofReal_natCast N, ENNReal.ofReal_rpow_of_nonneg (Nat.cast_nonneg _) (by positivity),
← ENNReal.ofReal_pow (by positivity : 0 ≤ K*192),
← ENNReal.ofReal_mul (by positivity : 0 ≤ (K*192)^d),
← ENNReal.ofReal_mul (by positivity : 0 ≤ H*r^β)]
congr 1
ring
end MemoryPrecision
end
end OAI
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